15.7 Triple integrals in Cylindrical Coordinates
Claudia Castro-Castro
Math 283 Spring 2020
Triple Integrals in Cylindrical Coordinates
- In 3D dimensions there is a coordinate system, called cylindrical coordinates, that is similar to polar coordinates and gives convenient descriptions of some commonly occurring surfaces and solids
- Suppose you want to find the total mass of the half cylinder with density function \( f(x,y,z) \)
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
- Suppose you want to find the total mass of the half cylinder with density function \( \rho(x,y,z) \) \[ mass=\iiint \limits_{\frac{1}{2} Cylinder} \rho(x,y,z)dV \]
- Projections
Recall: polar coordinates
- Cartesian coordinates \( (x, y) \) and polar coordinates \( (r,\theta) \)
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
-
Polar to rectangular \( (r,\theta)\rightarrow (x,y) \)
\[ x=r\cos \theta \\y=r\sin \theta \] -
Rectangular to polar \( (x,y)\rightarrow (r,\theta) \)
\[ r^2=x^2+y^2 \\ \tan \theta = \frac{y}{x},\;\cos \theta = \frac{x}{r},\;\sin \theta = \frac{y}{r} \]
Cylindrical coordinates
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
- A point \( P \) in 3-D space is represented by the ordered triple \[ (r, \theta , z) \]
- where \( r \) and \( \theta \) are polar coordinates of the projection of \( P \) onto the \( xy \)-plane
- and \( z \) is the directed distance from the \( xy \) -plane to \( P \)
-
Cylindrical to rectangular \( (r,\theta,z)\rightarrow (x,y,z) \)
\[ \begin{align} x=&r\cos \theta \\y=&r\sin \theta\\z=&z \end{align} \] -
Rectangular to cylindrical \( (x,y,z)\rightarrow (r,\theta,z) \)
\[ r=\sqrt{x^2+y^2} \\ \tan \theta = \frac{y}{x},\;\cos \theta = \frac{x}{r},\;\sin \theta = \frac{y}{r}\\z=z \]
Solids in Cylindrical coordinates
- We can re-define certain solids in Cylindrical coordinates.
Example: Cylinder
-
The cyircular cylinder with Cartesian equation \[ x^2+y^2=c^2 \]
- In cylindrical coordinates the surface of this cylinder has the equation \[ r=c \]
- and the solid cylinder as the set
\[ E=\left\{(r,\theta,z)|\; 0 \leq r\leq c,\; 0 \leq \theta \leq 2\pi,\; a\leq z\leq b \right\} \]
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
Solids in Cylindrical coordinates
Example: Cone
- The cyircular cone with Cartesian equation \[ z^2=x^2+y^2 \]
- Cone has radial symmetry
-
In cylindrical coordinates the surface of this cone has the equation \[ z=r \]
- and the solid cone as the set
\[ E=\{(r,\theta,z)|\; 0 \leq r\leq c,\; 0 \leq \theta \leq 2\pi,\; r \leq z\leq s \} \]
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
Evaluating Triple integrals in Cylindrical Coordinates
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
- Suppose that \( E \) is a type 1 solid with projection \( D \) onto, say, the \( xy \)-plane is conveniently described in polar coordinates
- We know that \[ \iiint_\limits{E} f(x,y,z)dV=\iint_\limits{D}\left[ \int_{\color{purple} {u_1(x,y)}}^{\color{purple} {u_2(x,y)}} f(x,y,z) {\color{purple} {dz}} \right] dA \]
- \( D \) is given in polar coordinates
- Convert a triple integral from rectangular to cylindrical coordinates using \[ \begin{align} x&=r\cos\theta\\ y&=r\sin\theta\\ z&=z\end{align} \]
-
\[ \iiint_\limits{E} f(r,\theta,z)dV_{cyl} = \iint_\limits{D}\left[ \int_{{\color{purple} {u_1(r,\theta)}}}^{{\color{purple} {u_2(r,\theta)}}} f(r,\theta,z) dz \right] dA_{polar} \]
- What is \( dV_{cyl} \)?
Volume element in Cylindrical Coordinates
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
- Volume element in cylindrical coordinates has volume
\[ dV_{cyl} =(Area\; of\; polar\;rectangle)(height) \] - \[ dV_{cyl} =(dA_{polar})(dz) \]
- \[ dV_{cyl} =(rdrd\theta)(dz) \]
-
\[ dV_{cyl} = r\: dz\: dr\:d\theta \]
Example cont'd
- A solid \( E \) lies within the cylinder \( x^2 + y^2 = 1 \), below the plane \( z = 4 \), and above the paraboloid \( z = 1 – x^2 – y^2 \). The density at any point is proportional to its distance from the axis of the cylinder. Find the total mass of \( E \).
- Sketch \( E \).
- Re-define the boundary surfaces in Cylindrical coordinates \[ r=\sqrt{x^2+y^2}\:\: or \:\: r^2=x^2+y^2 \]
- Cylinder \( x^2 + y^2 = 1 \) becomes \[ r = 1 \] \( \theta \in [0,2\pi] \)
- Paraboloid \( z= 1-x^2-y^2 \) becomes \( z = 1 – r^2 \)
- Plane \[ z= 4 \]
- Set \( E \) in cylindrical coordinates \[ E = \{(r,\theta,z)|\; 0\leq r \leq 1,\; 0 \leq \theta \leq 2\pi,\; 1-r^2 \leq z\leq 4 \} \]
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
Example cont'd
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
- Set up the total mass as a triple integral \[ mass=\int_{0}^{2\pi} \int_0^1 \int_{{\color{purple} {z=1-r^2}} }^{{\color{purple} {z=4}}} ( Kr )\; r\; {\color{purple} {dz}} \;dr\: d\theta \]
- Integrate with respect to \( z \) first \[ \begin{align} mass&=K \int_{0}^{2\pi} \int_0^1 \int_{{\color{purple} {z=1-r^2}} }^{{\color{purple} {z=4}}} r^2\; {\color{purple} {dz}} \;dr\: d\theta \\ &=K \int_{0}^{2\pi} \int_0^1 \left[r^2\: z\: \right]\bigg\vert_{{\color{purple} {z=1-r^2}} }^{{\color{purple} {z=4}}} \;dr\: d\theta\\ &=K \int_{0}^{2\pi} \int_0^1 r^2 \left[4-(1-r^2) \right] \;dr\: d\theta \end{align} \]
- Evaluate the double integral in polar coordinates \[ \begin{align} mass& = K \int_{0}^{2\pi} \int_0^1 \left(3r^2+r^4 \right) \;dr\: d\theta \end{align} \]
- Over a polar rectangle \[ \begin{align} mass&=K \int_{0}^{2\pi} d\theta \int_0^1 \left(3r^2+r^4 \right) \;dr\\ &= K\theta\bigg\vert_{\theta=0}^{\theta=2\pi} \left({r^3+\frac{r^5}{5}} \right)\bigg\vert_{r=0}^{r=1} = K(2\pi)\left( \frac{6}{5}\right) = \frac{12\pi K}{5} \end{align} \]
Example
-
Find the volume of a cone of base radius \( R \) and height \( H \) using cylindrical coordinates
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
- Volume triple integral \[ V(E)= \iiint \limits_E 1\;dV_{cyl} \]
- The cone \( E \) has a much simpler description in cylindrical coordinates: \[ E=\left\{(r,\theta,z)|\; 0\leq r \leq R,\; 0 \leq \theta \leq 2\pi,\; \frac{H}{R}r \leq z\leq H \right\} \]
- used radial symmetry to project to \( rz \)-plane
- Integrate with respect to \( z \) \[ V(E)= \int_0^{2\pi}\int_0^R \int_{z=\frac{H}{R}r}^{z=H} 1\;r\;dz\;dr\;d\theta \]
Example cont'd
- Integrate with respect to \( z \) \[ V(E)= \int_0^{2\pi}\int_0^R \left[\;r\;z \bigg\vert_{z=\frac{H}{R}r}^{z=H}\right] \;dr\;d\theta \]
- \[ V(E)= \int_0^{2\pi}\int_0^R \left(H r - \frac{H}{R} r^2 \right) \;dr\;d\theta \]
- Integrate over the polar rectangle \[ \begin{align} V(E)&= \int_0^{2\pi} d\theta \int_0^R \left(H r - \frac{H}{R} r^2 \right)\\ &=(2\pi) \left( H\frac{R^2}{2}-\frac{H}{R}\frac{R^3}{3}\right) \\ &=(2\pi)H\left( \frac{R^2}{2}-\frac{R^2}{3}\right)\\ &=(2\pi)HR^2\left( \frac{1}{2}-\frac{1}{3}\right)\\ &=(2\pi)HR^2\left(\frac{1}{6}\right)=\frac{\pi}{3} HR^2\end{align} \]